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</li> <li id="toc-Practical_usage" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Practical_usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Practical usage</span> </div> </a> <button aria-controls="toc-Practical_usage-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Practical usage subsection</span> </button> <ul id="toc-Practical_usage-sublist" class="vector-toc-list"> <li id="toc-Binary-coded_ternary" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary-coded_ternary"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Binary-coded ternary</span> </div> </a> <ul id="toc-Binary-coded_ternary-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tryte" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tryte"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Tryte</span> </div> </a> <ul id="toc-Tryte-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Sam-ch%C3%ACn-hoat" title="Sam-chìn-hoat – Minnan" lang="nan" hreflang="nan" data-title="Sam-chìn-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_ternari" title="Sistema ternari – Catalan" lang="ca" hreflang="ca" data-title="Sistema ternari" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Trojkov%C3%A1_soustava" title="Trojková soustava – Czech" lang="cs" hreflang="cs" data-title="Trojková soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tern%C3%A6rt_talsystem" title="Ternært talsystem – Danish" lang="da" hreflang="da" data-title="Ternært talsystem" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Tern%C3%A4rsystem" title="Ternärsystem – German" lang="de" hreflang="de" data-title="Ternärsystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%B9%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B1%CF%81%CE%AF%CE%B8%CE%BC%CE%B7%CF%83%CE%B7%CF%82" title="Τριαδικό σύστημα αρίθμησης – Greek" lang="el" hreflang="el" data-title="Τριαδικό σύστημα αρίθμησης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_ternario" title="Sistema ternario – Spanish" lang="es" hreflang="es" data-title="Sistema ternario" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Triuma_sistemo" title="Triuma sistemo – Esperanto" lang="eo" hreflang="eo" data-title="Triuma sistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_ternaire" title="Système ternaire – French" lang="fr" hreflang="fr" data-title="Système ternaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EC%A7%84%EB%B2%95" title="삼진법 – Korean" lang="ko" hreflang="ko" data-title="삼진법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ternara_nombrosistemo" title="Ternara nombrosistemo – Ido" lang="io" hreflang="io" data-title="Ternara nombrosistemo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_numerico_ternario" title="Sistema numerico ternario – Italian" lang="it" hreflang="it" data-title="Sistema numerico ternario" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Theluthi" title="Theluthi – Swahili" lang="sw" hreflang="sw" data-title="Theluthi" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_trin%C3%A8" title="Sistèm trinè – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm trinè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sistema_de_numera%C3%A7on_ternairo" title="Sistema de numeraçon ternairo – Mirandese" lang="mwl" hreflang="mwl" data-title="Sistema de numeraçon ternairo" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E9%80%B2%E6%B3%95" title="三進法 – Japanese" lang="ja" hreflang="ja" data-title="三進法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tern%C3%A6rt_talsystem" title="Ternært talsystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ternært talsystem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Uchlik_sanoq_sistemasi" title="Uchlik sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Uchlik sanoq sistemasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tr%C3%B3jkowy_system_liczbowy" title="Trójkowy system liczbowy – Polish" lang="pl" hreflang="pl" data-title="Trójkowy system liczbowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_numera%C3%A7%C3%A3o_tern%C3%A1rio" title="Sistema de numeração ternário – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de numeração ternário" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_ternar" title="Sistem ternar – Romanian" lang="ro" hreflang="ro" data-title="Sistem ternar" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Троичная система счисления – Russian" lang="ru" hreflang="ru" data-title="Троичная система счисления" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ternary_numeral_system" title="Ternary numeral system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ternary numeral system" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Lambar_Tubaxan" title="Lambar Tubaxan – Somali" lang="so" hreflang="so" data-title="Lambar Tubaxan" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%81%D1%82%D1%80%D1%83%D0%BA%D0%B8_%D0%BD%D1%83%D0%BC%D0%B5%D1%80%D0%B8%D1%87%D0%BA%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Троструки нумерички систем – Serbian" lang="sr" hreflang="sr" data-title="Троструки нумерички систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tern%C3%A4ra_talsystemet" title="Ternära talsystemet – Swedish" lang="sv" hreflang="sv" data-title="Ternära talsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%AA%E0%B8%B2%E0%B8%A1" title="เลขฐานสาม – Thai" lang="th" hreflang="th" data-title="เลขฐานสาม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7l%C3%BC_say%C4%B1_sistemi" title="Üçlü sayı sistemi – Turkish" lang="tr" hreflang="tr" data-title="Üçlü sayı sistemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D1%96%D0%B9%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Трійкова система числення – Ukrainian" lang="uk" hreflang="uk" data-title="Трійкова система числення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_tam_ph%C3%A2n" title="Hệ tam phân – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ tam phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E9%80%B2%E5%88%B6" title="三進制 – Cantonese" lang="yue" hreflang="yue" data-title="三進制" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%BF%9B%E5%88%B6" title="三进制 – Chinese" lang="zh" hreflang="zh" data-title="三进制" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1056486#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic</a></li> <li><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul> <hr /> <ul><li><a href="/wiki/Bengali_numerals" title="Bengali numerals">Bengali</a></li> <li><a href="/wiki/Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li> <li><a href="/wiki/Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li> <li><a href="/wiki/Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li> <li><a href="/wiki/Odia_numerals" title="Odia numerals">Odia</a></li> <li><a href="/wiki/Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li> <li><a href="/wiki/Tamil_numerals" title="Tamil numerals">Tamil</a></li> <li><a href="/wiki/Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li> <li><a href="/wiki/Telugu_script#Numerals" title="Telugu script">Telugu</a></li> <li><a href="/wiki/Kannada_script#Numerals" title="Kannada script">Kannada</a></li> <li><a href="/wiki/Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul> <hr /> <ul><li><a href="/wiki/Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li> <li><a href="/wiki/Balinese_numerals" title="Balinese numerals">Balinese</a></li> <li><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></li> <li><a href="/wiki/Javanese_numerals" title="Javanese numerals">Javanese</a></li> <li><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></li> <li><a href="/wiki/Lao_script#Numerals" title="Lao script">Lao</a></li> <li><a href="/wiki/Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li> <li><a href="/wiki/Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li> <li><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a> <ul><li><a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li> <li><a href="/wiki/Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li> <li><a href="/wiki/Japanese_numerals" title="Japanese numerals">Japanese</a></li> <li><a href="/wiki/Korean_numerals" title="Korean numerals">Korean</a></li> <li><a href="/wiki/Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul> <hr /> <dl><dt>Historic</dt></dl> <ul><li><a href="/wiki/Counting_rods" title="Counting rods">Counting rods</a></li> <li><a href="/wiki/Tangut_numerals" title="Tangut numerals">Tangut</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul> <hr /> <dl><dt><a href="/wiki/Ancient_history" title="Ancient history">Ancient</a></dt></dl> <ul><li><a href="/wiki/Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul> <hr /> <dl><dt><a href="/wiki/Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl> <ul><li><a href="/wiki/Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li> <li><a href="/wiki/Maya_numerals" title="Maya numerals">Mayan</a></li> <li><a href="/wiki/Muisca_numerals" title="Muisca numerals">Muisca</a></li> <li><a href="/wiki/Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li> <li><a href="/wiki/Quipu" title="Quipu">Quipu</a></li> <li><a href="/wiki/Rumi_Numeral_Symbols" title="Rumi Numeral Symbols">Rumi</a></li></ul> <hr /> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li> <li><a href="/wiki/Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</a> (Iñupiaq)</li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="/wiki/Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Common radices/bases</dt></dl> <ul><li><a href="/wiki/Binary_number" title="Binary number">2</a></li> <li><a class="mw-selflink selflink">3</a></li> <li><a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">4</a></li> <li><a href="/wiki/Quinary" title="Quinary">5</a></li> <li><a href="/wiki/Senary" title="Senary">6</a></li> <li><a href="/wiki/Octal" title="Octal">8</a></li> <li><a href="/wiki/Decimal" title="Decimal">10</a></li> <li><a href="/wiki/Duodecimal" title="Duodecimal">12</a></li> <li><a href="/wiki/Hexadecimal" title="Hexadecimal">16</a></li> <li><a href="/wiki/Vigesimal" title="Vigesimal">20</a></li> <li><a href="/wiki/Sexagesimal" title="Sexagesimal">60</a></li></ul> <hr /> <dl><dt><a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl> <ul><li><a href="/wiki/Bijective_numeration" title="Bijective numeration">Bijective</a><span class="nowrap"> </span>(<a href="/wiki/Unary_numeral_system" title="Unary numeral system">1</a>)</li> <li><a href="/wiki/Signed-digit_representation" title="Signed-digit representation">Signed-digit</a><span class="nowrap"> </span>(<a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li> <li><a href="/wiki/Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<a href="/wiki/Factorial_number_system" title="Factorial number system">factorial</a>)</li> <li><a href="/wiki/Negative_base" title="Negative base">Negative</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex</a><span class="nowrap"> </span>(<a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li> <li><a href="/wiki/Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<a href="/wiki/Golden_ratio_base" title="Golden ratio base">φ</a>)</li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></li></ul></div></div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Non-alphabetic</dt></dl> <ul><li><a href="/wiki/Aegean_numerals" title="Aegean numerals">Aegean</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic</a></li> <li><a href="/wiki/Aztec_script#Numerals" title="Aztec script">Aztec</a></li> <li><a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li> <li><a href="/wiki/Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li> <li><a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li> <li><a href="/wiki/Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li> <li><a href="/wiki/Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li> <li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric counting</a></li> <li><a href="/wiki/Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li> <li><a href="/wiki/Roman_numerals" title="Roman numerals">Roman</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li></ul> <hr /> <dl><dt><a href="/wiki/Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl> <ul><li><a href="/wiki/Abjad_numerals" title="Abjad numerals">Abjad</a></li> <li><a href="/wiki/Armenian_numerals" title="Armenian numerals">Armenian</a></li> <li><a href="/wiki/Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a> <ul><li><a href="/wiki/Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li> <li><a href="/wiki/Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li> <li><a href="/wiki/Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li> <li><a href="/wiki/Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li> <li><a href="/wiki/Georgian_numerals" title="Georgian numerals">Georgian</a></li> <li><a href="/wiki/Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek</a></li> <li><a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numeral_systems" title="Template:Numeral systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numeral_systems" title="Template talk:Numeral systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numeral_systems" title="Special:EditPage/Template:Numeral systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>ternary</b> <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'t' in 'tie'">t</span><span title="/ɜːr/: 'ur' in 'fur'">ɜːr</span><span title="'n' in 'nigh'">n</span><span title="/ər/: 'er' in 'letter'">ər</span><span title="/i/: 'y' in 'happy'">i</span></span>/</a></span></span> <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> (also called <b>base 3</b> or <b>trinary</b><sup id="cite_ref-Kindra2022_1-0" class="reference"><a href="#cite_note-Kindra2022-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) has <a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">three</a> as its <a href="/wiki/Radix" title="Radix">base</a>. Analogous to a <a href="/wiki/Bit" title="Bit">bit</a>, a ternary <a href="/wiki/Numerical_digit" title="Numerical digit">digit</a> is a <b>trit</b> (<b>tri</b>nary dig<b>it</b>). One trit is equivalent to <a href="/wiki/Binary_logarithm" title="Binary logarithm">log<sub>2</sub></a> 3 (about 1.58496) bits of <a href="/wiki/Units_of_information" title="Units of information">information</a>. </p><p>Although <i>ternary</i> most often refers to a system in which the three digits are all non–negative numbers; specifically <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a>, <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, and <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, the adjective also lends its name to the <a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a> system; comprising the digits <a href="/wiki/%E2%88%921" title="−1">−1</a>, 0 and +1, used in comparison logic and <a href="/wiki/Ternary_computer" title="Ternary computer">ternary computers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Comparison_to_other_bases">Comparison to other bases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=1" title="Edit section: Comparison to other bases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Representations of <a href="/wiki/Integer_number" class="mw-redirect" title="Integer number">integer numbers</a> in ternary do not get uncomfortably lengthy as quickly as in <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a>. For example, <a href="/wiki/Decimal" title="Decimal">decimal</a> <a href="/wiki/365_(number)" title="365 (number)">365</a><sub>(10)</sub> or <a href="/wiki/Senary" title="Senary">senary</a> <span style="white-space:nowrap">1<span style="margin-left:0.25em">405</span></span><sub>(6)</sub> corresponds to binary <span style="white-space:nowrap">1<span style="margin-left:0.25em">0110</span><span style="margin-left:0.25em">1101</span></span><sub>(2)</sub> (nine <a href="/wiki/Bit" title="Bit">bits</a>) and to ternary <span style="white-space:nowrap">111<span style="margin-left:0.25em">112</span></span><sub>(3)</sub> (six digits). However, they are still far less compact than the corresponding representations in bases such as <a href="/wiki/Decimal" title="Decimal">decimal</a> – see below for a compact way to codify ternary using nonary (base 9) and <a href="/wiki/Septemvigesimal" class="mw-redirect" title="Septemvigesimal">septemvigesimal</a> (base 27). </p> <table class="wikitable" style="float:right; text-align:center"> <caption>A ternary <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a> </caption> <tbody><tr> <th>×</th> <th><b>1</b></th> <th><b>2</b></th> <th><b>10</b></th> <th><b>11</b></th> <th><b>12</b></th> <th><b>20</b></th> <th><b>21</b></th> <th><b>22</b></th> <th><b>100</b> </th></tr> <tr> <th><b>1</b> </th> <td>1</td> <td>2</td> <td>10</td> <td>11</td> <td>12</td> <td>20</td> <td>21</td> <td>22</td> <td>100 </td></tr> <tr> <th><b>2</b> </th> <td>2</td> <td>11</td> <td>20</td> <td>22</td> <td>101</td> <td>110</td> <td>112</td> <td>121</td> <td>200 </td></tr> <tr> <th><b>10</b> </th> <td>10</td> <td>20</td> <td>100</td> <td>110</td> <td>120</td> <td>200</td> <td>210</td> <td>220</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span></span> </td></tr> <tr> <th><b>11</b> </th> <td>11</td> <td>22</td> <td>110</td> <td>121</td> <td>202</td> <td>220</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">001</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">012</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">100</span></span> </td></tr> <tr> <th><b>12</b> </th> <td>12</td> <td>101</td> <td>120</td> <td>202</td> <td>221</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">010</span></span> </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">022</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">111</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">200</span></span> </td></tr> <tr> <th><b>20</b> </th> <td>20</td> <td>110</td> <td>200</td> <td>220</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">010</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">100</span></span> </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">120</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">210</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">000</span></span> </td></tr> <tr> <th><b>21</b> </th> <td>21</td> <td>112</td> <td>210</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">001</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">022</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">120</span></span> </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">211</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">002</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">100</span></span> </td></tr> <tr> <th><b>22</b> </th> <td>22</td> <td>121</td> <td>220</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">012</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">111</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">210</span></span> </td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">002</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">101</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">200</span></span> </td></tr> <tr> <th><b>100</b> </th> <td>100</td> <td>200</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">100</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">200</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">000</span></span> </td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">100</span></span></td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">200</span></span></td> <td><span style="white-space:nowrap">10<span style="margin-left:0.25em">000</span></span> </td></tr></tbody></table> <dl><dd><table class="wikitable"> <caption><b>Numbers from 0 to 3<sup>3</sup> − 1 in standard ternary</b> </caption> <tbody><tr align="center"> <th>Ternary </th> <td>0</td> <td>1</td> <td>2</td> <td>10</td> <td>11</td> <td>12</td> <td>20</td> <td>21</td> <td>22 </td></tr> <tr align="center"> <th>Binary </th> <td>0</td> <td>1</td> <td>10</td> <td>11</td> <td>100</td> <td>101</td> <td>110</td> <td>111</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span></span> </td></tr> <tr align="center"> <th>Senary </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>10</td> <td>11</td> <td>12 </td></tr> <tr align="center"> <th>Decimal </th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8 </th></tr> <tr> <td colspan="10" style="background-color:white;"> </td></tr> <tr align="center"> <th>Ternary </th> <td>100</td> <td>101</td> <td>102</td> <td>110</td> <td>111</td> <td>112</td> <td>120</td> <td>121</td> <td>122 </td></tr> <tr align="center"> <th>Binary </th> <td>1001</td> <td>1010</td> <td>1011</td> <td>1100</td> <td>1101</td> <td>1110</td> <td>1111 </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0000</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0001</span></span> </td></tr> <tr align="center"> <th>Senary </th> <td>13</td> <td>14</td> <td>15</td> <td>20</td> <td>21</td> <td>22</td> <td>23</td> <td>24</td> <td>25 </td></tr> <tr align="center"> <th>Decimal </th> <th>9</th> <th>10</th> <th>11</th> <th>12</th> <th>13</th> <th>14</th> <th>15</th> <th>16</th> <th>17 </th></tr> <tr> <td colspan="10" style="background-color:white;"> </td></tr> <tr align="center"> <th>Ternary </th> <td>200</td> <td>201</td> <td>202</td> <td>210</td> <td>211</td> <td>212</td> <td>220</td> <td>221</td> <td>222 </td></tr> <tr align="center"> <th>Binary </th> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0010</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0011</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0100</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0101</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0110</span></span> </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">0111</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">1000</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">1001</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">1010</span></span> </td></tr> <tr align="center"> <th>Senary </th> <td>30</td> <td>31</td> <td>32</td> <td>33</td> <td>34</td> <td>35</td> <td>40</td> <td>41</td> <td>42 </td></tr> <tr align="center"> <th>Decimal </th> <th>18</th> <th>19</th> <th>20</th> <th>21</th> <th>22</th> <th>23</th> <th>24</th> <th>25</th> <th>26 </th></tr></tbody></table></dd></dl> <dl><dd></dd></dl> <dl><dd><table class="wikitable"> <caption><b>Powers of 3 in ternary</b> </caption> <tbody><tr align="center"> <th>Ternary </th> <td>1</td> <td>10</td> <td>100</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span></span></td> <td><span style="white-space:nowrap">10<span style="margin-left:0.25em">000</span></span> </td></tr> <tr align="center"> <th>Binary </th> <td>1</td> <td>11</td> <td>1001</td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">1011</span></span></td> <td><span style="white-space:nowrap">101<span style="margin-left:0.25em">0001</span></span> </td></tr> <tr align="center"> <th>Senary </th> <td>1</td> <td>3</td> <td>13</td> <td>43</td> <td>213 </td></tr> <tr align="center"> <th>Decimal </th> <td>1</td> <td>3</td> <td>9</td> <td>27</td> <td>81 </td></tr> <tr align="center"> <th>Power </th> <th><span style="font-size:120%">3</span><sup>0</sup></th> <th><span style="font-size:120%">3</span><sup>1</sup></th> <th><span style="font-size:120%">3</span><sup>2</sup> </th> <th><span style="font-size:120%">3</span><sup>3</sup></th> <th><span style="font-size:120%">3</span><sup>4</sup> </th></tr> <tr> <td colspan="10" style="background-color:white;"> </td></tr> <tr align="center"> <th>Ternary </th> <td><span style="white-space:nowrap">100<span style="margin-left:0.25em">000</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span></td> <td><span style="white-space:nowrap">10<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td><span style="white-space:nowrap">100<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span></td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td></tr> <tr align="center"> <th>Binary </th> <td><span style="white-space:nowrap">1111<span style="margin-left:0.25em">0011</span></span></td> <td><span style="white-space:nowrap">10<span style="margin-left:0.25em">1101</span><span style="margin-left:0.25em">1001</span></span></td> <td><span style="white-space:nowrap">1000<span style="margin-left:0.25em">1000</span><span style="margin-left:0.25em">1011</span></span> </td> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">1001</span><span style="margin-left:0.25em">1010</span><span style="margin-left:0.25em">0001</span></span></td> <td><span style="white-space:nowrap">100<span style="margin-left:0.25em">1100</span><span style="margin-left:0.25em">1110</span><span style="margin-left:0.25em">0011</span></span> </td></tr> <tr align="center"> <th>Senary </th> <td><span style="white-space:nowrap">1<span style="margin-left:0.25em">043</span></span></td> <td><span style="white-space:nowrap">3<span style="margin-left:0.25em">213</span></span></td> <td><span style="white-space:nowrap">14<span style="margin-left:0.25em">043</span></span></td> <td><span style="white-space:nowrap">50<span style="margin-left:0.25em">213</span></span></td> <td><span style="white-space:nowrap">231<span style="margin-left:0.25em">043</span></span> </td></tr> <tr align="center"> <th>Decimal </th> <td>243</td> <td>729</td> <td><span style="white-space:nowrap">2<span style="margin-left:0.25em">187</span></span></td> <td><span style="white-space:nowrap">6<span style="margin-left:0.25em">561</span></span></td> <td><span style="white-space:nowrap">19<span style="margin-left:0.25em">683</span></span> </td></tr> <tr align="center"> <th>Power </th> <th><span style="font-size:120%">3</span><sup>5</sup></th> <th><span style="font-size:120%">3</span><sup>6</sup></th> <th><span style="font-size:120%">3</span><sup>7</sup> </th> <th><span style="font-size:120%">3</span><sup>8</sup></th> <th><span style="font-size:120%">3</span><sup>9</sup> </th></tr></tbody></table></dd></dl> <p>As for <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, ternary offers a convenient way to represent <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> as same as senary (as opposed to its cumbersome representation as an infinite string of <a href="/wiki/Recurring_decimal" class="mw-redirect" title="Recurring decimal">recurring digits</a> in decimal); but a major drawback is that, in turn, ternary does not offer a finite representation for <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> (nor for <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span>, etc.), because <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a> is not a <a href="/wiki/Prime_number" title="Prime number">prime</a> <a href="/wiki/Factorization" title="Factorization">factor</a> of the base; as with base two, one-tenth (decimal<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span>, senary <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">14</span></span>⁠</span>) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span>, decimal <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>). </p> <dl><dd><table class="wikitable"> <caption><b>Fractions in ternary</b> </caption> <tbody><tr align="center"> <th>Fraction </th> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">11</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></b></td> <td><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">13</span></span>⁠</span></b> </td></tr> <tr align="center"> <th>Ternary </th> <td>0.<span style="text-decoration:overline;">1</span></td> <td>0.1</td> <td>0.<span style="text-decoration:overline;">02</span></td> <td>0.<span style="text-decoration:overline;">0121</span></td> <td>0.0<span style="text-decoration:overline;">1</span></td> <td>0.<span style="text-decoration:overline;">010212</span></td> <td>0.<span style="text-decoration:overline;">01</span></td> <td>0.01</td> <td>0.<span style="text-decoration:overline;">0022</span></td> <td>0.<span style="text-decoration:overline;">00211</span></td> <td>0.0<span style="text-decoration:overline;">02</span></td> <td>0.<span style="text-decoration:overline;">002</span> </td></tr> <tr align="center"> <th>Binary </th> <td>0.1</td> <td>0.<span style="text-decoration:overline;">01</span></td> <td>0.01</td> <td>0.<span style="text-decoration:overline;">0011</span></td> <td>0.0<span style="text-decoration:overline;">01</span></td> <td>0.<span style="text-decoration:overline;">001</span></td> <td>0.001</td> <td>0.<span style="text-decoration:overline;">000111</span></td> <td>0.0<span style="text-decoration:overline;">0011</span></td> <td>0.<span style="text-decoration:overline;">0001011101</span></td> <td>0.00<span style="text-decoration:overline;">01</span></td> <td>0.<span style="text-decoration:overline;">000100111011</span> </td></tr> <tr align="center"> <th>Senary </th> <td>0.3</td> <td>0.2</td> <td>0.13</td> <td>0.<span style="text-decoration:overline;">1</span></td> <td>0.1</td> <td>0.<span style="text-decoration:overline;">05</span></td> <td>0.043</td> <td>0.04</td> <td>0.0<span style="text-decoration:overline;">3</span></td> <td>0.<span style="text-decoration:overline;">0313452421</span></td> <td>0.03</td> <td>0.<span style="text-decoration:overline;">024340531215</span> </td></tr> <tr align="center"> <th>Decimal </th> <th>0.5</th> <th>0.<span style="text-decoration:overline;">3</span></th> <th>0.25</th> <th>0.2</th> <th>0.1<span style="text-decoration:overline;">6</span></th> <th>0.<span style="text-decoration:overline;">142857</span></th> <th>0.125 </th> <th>0.<span style="text-decoration:overline;">1</span></th> <th>0.1</th> <th>0.<span style="text-decoration:overline;">09</span></th> <th>0.08<span style="text-decoration:overline;">3</span></th> <th>0.<span style="text-decoration:overline;">076923</span> </th></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sum_of_the_digits_in_ternary_as_opposed_to_binary">Sum of the digits in ternary as opposed to binary</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=2" title="Edit section: Sum of the digits in ternary as opposed to binary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value of a binary number with <i>n</i> bits that are all 1 is <span class="texhtml">2<sup><i>n</i></sup> − 1</span>. </p><p>Similarly, for a number <i>N</i>(<i>b</i>, <i>d</i>) with base <i>b</i> and <i>d</i> digits, all of which are the maximal digit value <span class="texhtml"><i>b</i> − 1</span>, we can write: </p> <dl><dd><span class="texhtml"><i>N</i>(<i>b</i>, <i>d</i>) = (<i>b</i> − 1)<i>b</i><sup><i>d</i>−1</sup> + (<i>b</i> − 1)<i>b</i><sup><i>d</i>−2</sup> + … + (<i>b</i> − 1)<i>b</i><sup>1</sup> + (<i>b</i> − 1)<i>b</i><sup>0</sup>,</span></dd> <dd><span class="texhtml"><span style="color:white"><i>N</i>(<i>b</i>, <i>d</i>)</span> = (<i>b</i> − 1)(<i>b</i><sup><i>d</i>−1</sup> + <i>b</i><sup><i>d</i>−2</sup> + … + <i>b</i><sup>1</sup> + 1),</span></dd> <dd><span class="texhtml"><span style="color:white"><i>N</i>(<i>b</i>, <i>d</i>)</span> = (<i>b</i> − 1)<i>M</i></span>.</dd> <dd><span class="texhtml"><i>bM</i> = <i>b</i><sup><i>d</i></sup> + <i>b</i><sup><i>d</i>−1</sup> + … + <i>b</i><sup>2</sup> + <i>b</i><sup>1</sup></span> and</dd> <dd><span class="texhtml">−<i>M</i> = −<i>b</i><sup><i>d</i>−1</sup> − <i>b</i><sup><i>d</i>−2</sup> − ... − b<sup>1</sup> − 1</span>, so</dd> <dd><span class="texhtml"><i>bM</i> − <i>M</i> = <i>b</i><sup><i>d</i></sup> − 1</span>, or</dd> <dd><span class="texhtml"><i>M</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i><sup><i>d</i></sup> − 1</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span>.</span></dd></dl> <p>Then </p> <dl><dd><span class="texhtml"><i>N</i>(<i>b</i>, <i>d</i>) = (<i>b</i> − 1)<i>M</i>,</span></dd> <dd><span class="texhtml"><span style="color:white"><i>N</i>(<i>b</i>, <i>d</i>)</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">(<i>b</i> − 1)(<i>b</i><sup><i>d</i></sup> − 1)</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span>,</span></dd> <dd><span class="texhtml"><span style="color:white"><i>N</i>(<i>b</i>, <i>d</i>)</span> = <i>b</i><sup><i>d</i></sup> − 1.</span></dd></dl> <p>For a three-digit ternary number, <span class="texhtml"><i>N</i>(3, 3) = 3<sup>3</sup> − 1 = 26 = 2 × 3<sup>2</sup> + 2 × 3<sup>1</sup> + 2 × 3<sup>0</sup> = 18 + 6 + 2</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Compact_ternary_representation:_base_9_and_27">Compact ternary representation: base 9 and 27</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=3" title="Edit section: Compact ternary representation: base 9 and 27"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nonary (base 9, each digit is two ternary digits) or <a href="/wiki/Septemvigesimal" class="mw-redirect" title="Septemvigesimal">septemvigesimal</a> (base 27, each digit is three ternary digits) can be used for compact representation of ternary, similar to how <a href="/wiki/Octal" title="Octal">octal</a> and <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> systems are used in place of <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_usage">Practical usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=4" title="Edit section: Practical usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fewest_weights_balance_puzzle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Fewest_weights_balance_puzzle.svg/220px-Fewest_weights_balance_puzzle.svg.png" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Fewest_weights_balance_puzzle.svg/330px-Fewest_weights_balance_puzzle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Fewest_weights_balance_puzzle.svg/440px-Fewest_weights_balance_puzzle.svg.png 2x" data-file-width="512" data-file-height="768" /></a><figcaption>Use of ternary numbers to balance an unknown integer weight from 1 to 40 kg with weights of 1, 3, 9 and 27 kg (4 ternary digits actually gives 3<sup>4</sup> = 81 possible combinations: −40 to +40, but only the positive values are useful)</figcaption></figure> <p>In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in <a href="/wiki/CMOS" title="CMOS">CMOS</a> circuits, and also in <a href="/wiki/Transistor%E2%80%93transistor_logic" title="Transistor–transistor logic">transistor–transistor logic</a> with <a href="/wiki/Push%E2%80%93pull_output" title="Push–pull output">totem-pole output</a>. The output is said to either be low (<a href="/wiki/Ground_(electricity)" title="Ground (electricity)">grounded</a>), high, or open (<a href="/wiki/High_impedance" title="High impedance">high-<i>Z</i></a>). In this configuration the output of the circuit is actually not connected to any <a href="/wiki/Voltage" title="Voltage">voltage</a> reference at all. Where the signal is usually grounded to a certain reference, or at a certain voltage level, the state is said to be high <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> because it is open and serves its own reference. Thus, the actual voltage level is sometimes unpredictable. </p><p>A rare "ternary point" in common use is for defensive statistics in American <a href="/wiki/Baseball" title="Baseball">baseball</a> (usually just for <a href="/wiki/Pitcher" title="Pitcher">pitchers</a>), to denote fractional parts of an inning. Since the team on offense is allowed three <a href="/wiki/Out_(baseball)" title="Out (baseball)">outs</a>, each out is considered one third of a defensive inning and is denoted as <b>.1</b>. For example, if a player pitched all of the 4th, 5th and 6th innings, plus achieving 2 outs in the 7th inning, his <a href="/wiki/Innings_pitched" title="Innings pitched">innings pitched</a> column for that game would be listed as <b>3.2</b>, the equivalent of <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac">3<span class="sr-only">+</span><span class="num">2</span>⁄<span class="den">3</span></span> (which is sometimes used as an alternative by some record keepers). In this usage, only the fractional part of the number is written in ternary form.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Ternary numbers can be used to convey self-similar structures like the <a href="/wiki/Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpinski triangle</a> or the <a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a> conveniently. Additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have a ternary expression that does not contain any instance of the digit 1.<sup id="cite_ref-Soltanifar_2006_1_4-0" class="reference"><a href="#cite_note-Soltanifar_2006_1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Soltanifar_2006_2_5-0" class="reference"><a href="#cite_note-Soltanifar_2006_2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Any terminating expansion in the ternary system is equivalent to the expression that is identical up to the term preceding the last non-zero term followed by the term one less than the last non-zero term of the first expression, followed by an infinite tail of twos. For example: 0.1020 is equivalent to 0.1012222... because the expansions are the same until the "two" of the first expression, the two was decremented in the second expansion, and trailing zeros were replaced with trailing twos in the second expression. </p><p>Ternary is the integer base with the lowest <a href="/wiki/Radix_economy" class="mw-redirect" title="Radix economy">radix economy</a>, followed closely by <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> and <a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">quaternary</a>. This is due to its proximity to the <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a> <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><i>e</i></a>. It has been used for some computing systems because of this efficiency. It is also used to represent three-option <i>trees</i>, such as phone menu systems, which allow a simple path to any branch. </p><p>A form of <a href="/wiki/Redundant_binary_representation" title="Redundant binary representation">redundant binary representation</a> called a binary signed-digit number system, a form of <a href="/wiki/Signed-digit_representation" title="Signed-digit representation">signed-digit representation</a>, is sometimes used in low-level software and hardware to accomplish fast addition of integers because it can eliminate <a href="/wiki/Carry_(arithmetic)" title="Carry (arithmetic)">carries</a>.<sup id="cite_ref-Phatak_1994_6-0" class="reference"><a href="#cite_note-Phatak_1994-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Binary-coded_ternary">Binary-coded ternary</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=5" title="Edit section: Binary-coded ternary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Simulation of ternary computers using binary computers, or interfacing between ternary and binary computers, can involve use of binary-coded ternary (BCT) numbers, with two or three bits used to encode each trit.<sup id="cite_ref-Frieder_1975_7-0" class="reference"><a href="#cite_note-Frieder_1975-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Parhami_2013_8-0" class="reference"><a href="#cite_note-Parhami_2013-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> BCT encoding is analogous to <a href="/wiki/Binary-coded_decimal" title="Binary-coded decimal">binary-coded decimal</a> (BCD) encoding. If the trit values 0, 1 and 2 are encoded 00, 01 and 10, conversion in either direction between binary-coded ternary and binary can be done in <a href="/wiki/Time_complexity#Logarithmic_time" title="Time complexity">logarithmic time</a>.<sup id="cite_ref-Jones_2016_1_9-0" class="reference"><a href="#cite_note-Jones_2016_1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> A library of <a href="/wiki/C_(programming_language)" title="C (programming language)">C code</a> supporting BCT arithmetic is available.<sup id="cite_ref-Jones_2016_2_10-0" class="reference"><a href="#cite_note-Jones_2016_2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tryte">Tryte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=6" title="Edit section: Tryte"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some <a href="/wiki/Ternary_computer" title="Ternary computer">ternary computers</a> such as the <a href="/wiki/Setun" title="Setun">Setun</a> defined a <b>tryte</b> to be six trits<sup id="cite_ref-Impagliazzo_2006_11-0" class="reference"><a href="#cite_note-Impagliazzo_2006-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> or approximately 9.5 <a href="/wiki/Bit" title="Bit">bits</a> (holding more information than the <i>de facto</i> <a href="/wiki/Binary_number" title="Binary number">binary</a> <a href="/wiki/Byte" title="Byte">byte</a>).<sup id="cite_ref-Brousentsov_2010_12-0" class="reference"><a href="#cite_note-Brousentsov_2010-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Qutrit" title="Qutrit">Qutrit</a></li> <li><a href="/wiki/Setun" title="Setun">Setun</a>, a <a href="/wiki/Ternary_computer" title="Ternary computer">ternary computer</a></li> <li><a href="/wiki/Ternary_logic" class="mw-redirect" title="Ternary logic">Ternary logic</a></li> <li><i><a href="/wiki/Taixuanjing" title="Taixuanjing">Taixuanjing</a></i></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output 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(2015-12-29). <a rel="nofollow" class="external text" href="http://www.cs.uiowa.edu/~jones/ternary/libtern.shtml">"Ternary Data Types for C Programmers"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ternary+Data+Types+for+C+Programmers&rft.date=2015-12-29&rft.aulast=Jones&rft.aufirst=Douglas+W.&rft_id=http%3A%2F%2Fwww.cs.uiowa.edu%2F~jones%2Fternary%2Flibtern.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+numeral+system" class="Z3988"></span></span> </li> <li id="cite_note-Impagliazzo_2006-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Impagliazzo_2006_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFImpagliazzoProydakov2006" class="citation conference cs1">Impagliazzo, John; Proydakov, Eduard (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-jSqCAAAQBAJ"><i>Perspectives on Soviet and Russian Computing</i></a>. First IFIP WG 9.7 Conference, SoRuCom 2006. Petrozavodsk, Russia: <a href="/wiki/Springer_(publisher)" class="mw-redirect" title="Springer (publisher)">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-64222816-2" title="Special:BookSources/978-3-64222816-2"><bdi>978-3-64222816-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Perspectives+on+Soviet+and+Russian+Computing&rft.place=Petrozavodsk%2C+Russia&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-64222816-2&rft.aulast=Impagliazzo&rft.aufirst=John&rft.au=Proydakov%2C+Eduard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-jSqCAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+numeral+system" class="Z3988"></span></span> </li> <li id="cite_note-Brousentsov_2010-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brousentsov_2010_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrousentsovMaslovRamil_AlvarezZhogolev" class="citation web cs1">Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E. A. <a rel="nofollow" class="external text" href="http://www.computer-museum.ru/english/setun.htm">"Development of ternary computers at Moscow State University"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2010-01-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Development+of+ternary+computers+at+Moscow+State+University&rft.aulast=Brousentsov&rft.aufirst=N.+P.&rft.au=Maslov%2C+S.+P.&rft.au=Ramil+Alvarez%2C+J.&rft.au=Zhogolev%2C+E.+A.&rft_id=http%3A%2F%2Fwww.computer-museum.ru%2Fenglish%2Fsetun.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+numeral+system" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayes2001" class="citation journal cs1"><a href="/wiki/Brian_Hayes_(scientist)" title="Brian Hayes (scientist)">Hayes, Brian</a> (November–December 2001). <a rel="nofollow" class="external text" href="http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf">"Third base"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/American_Scientist" title="American Scientist">American Scientist</a></i>. <b>89</b> (6). <a href="/wiki/Sigma_Xi" title="Sigma Xi">Sigma Xi</a>, the Scientific Research Society: 490–494. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2001.40.3268">10.1511/2001.40.3268</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191030114823/http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2019-10-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-04-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=Third+base&rft.volume=89&rft.issue=6&rft.pages=490-494&rft.date=2001-11%2F2001-12&rft_id=info%3Adoi%2F10.1511%2F2001.40.3268&rft.aulast=Hayes&rft.aufirst=Brian&rft_id=http%3A%2F%2Fbit-player.org%2Fwp-content%2Fextras%2Fbph-publications%2FAmSci-2001-11-Hayes-ternary.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+numeral+system" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_numeral_system&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.washingtonart.net/whealton/ternary.html">Ternary Arithmetic</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514130533/http://www.washingtonart.net/whealton/ternary.html">Archived</a> 2011-05-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.mortati.com/glusker/fowler/index.htm">The ternary calculating machine of Thomas Fowler</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathsisfun.com/numbers/convert-base.php?to=ternary">Ternary Base Conversion</a> – includes fractional part, from Maths Is Fun</li> <li><a rel="nofollow" class="external text" href="http://www.americanscientist.org/issues/pub/third-base/3">Gideon Frieder's replacement ternary numeral system</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output 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